Ecological Economics 101 (2014) 90–102
Contents lists available at ScienceDirect
Ecological Economics
journal homepage: www.elsevier.com/locate/ecolecon
Methodological and Ideological Options
Human and nature dynamics (HANDY): Modeling inequality and use of
resources in the collapse or sustainability of societies
Safa Motesharrei a,⁎, Jorge Rivas b, Eugenia Kalnay c
a
b
c
School of Public Policy and Department of Mathematics, University of Maryland; and National Socio-Environmental Synthesis Center (SESYNC)
Department of Political Science, University of Minnesota; and Institute of Global Environment and Society (IGES)
Department of Atmospheric and Oceanic Science and Institute of Physical Science and Technology, University of Maryland
a r t i c l e
i n f o
Article history:
Received 1 October 2012
Received in revised form 11 February 2014
Accepted 18 February 2014
Available online 2 April 2014
Keywords:
Human–nature dynamics
Societal collapse
Carrying capacity
Overshoot vs. sustainability
Economic inequality
Ecological strain
a b s t r a c t
There are widespread concerns that current trends in resource-use are unsustainable, but possibilities of
overshoot/collapse remain controversial. Collapses have occurred frequently in history, often followed by centuries of economic, intellectual, and population decline. Many different natural and social phenomena have been
invoked to explain specific collapses, but a general explanation remains elusive.
In this paper, we build a human population dynamics model by adding accumulated wealth and economic inequality to a predator–prey model of humans and nature. The model structure, and simulated scenarios that
offer significant implications, are explained. Four equations describe the evolution of Elites, Commoners, Nature,
and Wealth. The model shows Economic Stratification or Ecological Strain can independently lead to collapse, in
agreement with the historical record.
The measure “Carrying Capacity” is developed and its estimation is shown to be a practical means for early detection of a collapse. Mechanisms leading to two types of collapses are discussed. The new dynamics of this model
can also reproduce the irreversible collapses found in history. Collapse can be avoided, and population can reach a
steady state at maximum carrying capacity if the rate of depletion of nature is reduced to a sustainable level and if
resources are distributed equitably.
© 2014 The Authors. Published by Elsevier B.V. Open access under CC BY license.
1. Introduction
There are widespread concerns that current trends in population
and resource-use are unsustainable, but the possibilities of an overshoot
and collapse remain unclear and controversial. How real is the possibility of a societal collapse? Can complex, advanced civilizations really
collapse? It is common to portray human history as a relentless and inevitable trend toward greater levels of social complexity, political organization, and economic specialization, with the development of more
complex and capable technologies supporting ever-growing population, all sustained by the mobilization of ever-increasing quantities of
material, energy, and information. Yet this is not inevitable. In fact,
cases where this seemingly near-universal, long-term trend has been
severely disrupted by a precipitous collapse – often lasting centuries –
have been quite common. A brief review of some examples of collapses
suggests that the process of rise-and-collapse is actually a recurrent
cycle found throughout history, making it important to establish a
⁎ Corresponding author.
E-mail addresses: ssm@umd.edu (S. Motesharrei), jorgerodrigorivas@gmail.com
(J. Rivas), ekalnay@atmos.umd.edu (E. Kalnay).
http://dx.doi.org/10.1016/j.ecolecon.2014.02.014
0921-8009 © 2014 The Authors. Published by Elsevier B.V. Open access under CC BY license.
general explanation of this process (Chase-Dunn and Hall, 1997; Goldstein, 1988; Meadows et al., 1972; Modelski, 1987; Tainter, 1988;
Turchin and Nefedov, 2009; Yoffee and Cowgill, 1988).
The Roman Empire's dramatic collapse (followed by many centuries
of population decline, economic deterioration, intellectual regression,
and the disappearance of literacy) is well known, but it was not the
first rise-and-collapse cycle in Europe. Prior to the rise of Classical
Greco-Roman civilization, both the Minoan and Mycenaean Civilizations had each risen, reached very advanced levels of civilization, and
then collapsed virtually completely (Morris, 2006; Redman, 1999).
The history of Mesopotamia – the very cradle of civilization, agriculture,
complex society, and urban life – presents a series of rise-and-declines including the Sumerians, the Akkadian, Assyrian, Babylonian, Achaemenid,
Seleucid, Parthian, Sassanid, Umayyad, and Abbasid Empires (Redman
et al., 2004; Yoffee, 1979). In neighboring Egypt, this cycle also appeared
repeatedly. In both Anatolia and in the Indus Valley, the very large and
long-lasting Hittite and Harrapan civilizations both collapsed so
completely that their very existence was unknown until modern archeology rediscovered them. Similar cycles of rise and collapse occurred repeatedly in India, most notably with the Mauryan and the Gupta
Empires (Edwards et al., 1971, 1973; Jansen et al., 1991; Kenoyer,
1998; Thapar, 2004). Southeast Asia similarly experienced “multiple
S. Motesharrei et al. / Ecological Economics 101 (2014) 90–102
and overlapping histories of collapse and regeneration” over 15 centuries, culminating in the Khmer Empire based in Angkor, which itself
was depopulated and swallowed by the forest during the 15th Century
(Stark, 2006). Chinese history is, very much like Egypt's, full of repeated
cycles of rises and collapses, with each of the Zhou, Han, Tang, and Song
Empires followed by a very serious collapse of political authority and socioeconomic progress (Chu and Lee, 1994; Lee, 1931; Needham and
Wang, 1956).
Collapses are not restricted to the “Old World”. The collapse of Maya
Civilization is well known and evokes widespread fascination, both because of the advanced nature of Mayan society and because of the depth
of the collapse (Demerest et al., 2004; Webster, 2002). As Diamond
(2005) puts it, it is difficult to ignore “the disappearance of between
90 and 99% of the Maya population after A.D. 800 …and the disappearance of kings, Long Count calendars, and other complex political and
cultural institutions.” In the nearby central highlands of Mexico, a
number of powerful states also rose to high levels of power and prosperity and then rapidly collapsed, Teotihuacan (the sixth largest city in the
world in the 7th C) and Monte Alban being just the largest of these to
experience dramatic collapse, with their populations declining to
about 20–25% of their peak within just a few generations (Tainter,
1988).
We know of many other collapses including Mississippian Cultures
such as Cahokia, South West US cultures such as the Pueblo and
Hohokam, Andean civilizations such as Tiwanaku, Sub-Saharan civilizations such as Great Zimbabwe, and many collapses across the Pacific
Islands, such as Easter Island. It is also likely other collapses have also
occurred in societies that were not at a sufficient level of complexity
to produce written records or archeological evidence. Indeed, a recent
study (Shennan et al., 2013) of the Neolithic period in Europe has
shown that “in contrast to the steady population growth usually assumed, the introduction of agriculture into Europe was followed by a
boom-and-bust pattern in the density of regional populations”. Furthermore “most regions show more than one boom–bust pattern”, and in
most regions, population declines “of the order of the 30–60%” can be
found. The authors also argue that, rather than climate change or
diseases, the timing and evidence point to endogenous causes for
these collapses in 19 out of 23 cases studied, suggesting the possibility of “rapid population growth driven by farming to unsustainable
levels”. Moreover, through wavelet analysis of the archeological data,
S. Downey [personal communication] has shown that the average
length of such boom-and-bust cycles is about 300–500 years.
In summary, despite the common impression that societal collapse is
rare, or even largely fictional, “the picture that emerges is of a process
recurrent in history, and global in its distribution” (Tainter, 1988). See
also Yoffee and Cowgill (1988), Goldstein (1988), Ibn Khaldun (1958),
Kondratieff (1984), and Parsons (1991). As Turchin and Nefedov
(2009) contend, there is a great deal of support for “the hypothesis
that secular cycles — demographic–social–political oscillations of a
very long period (centuries long) are the rule, rather than an exception
in the large agrarian states and empires.”
This brings up the question of whether modern civilization is similarly susceptible. It may seem reasonable to believe that modern civilization, armed with its greater technological capacity, scientific
knowledge, and energy resources, will be able to survive and endure
whatever crises historical societies succumbed to. But the brief overview of collapses demonstrates not only the ubiquity of the phenomenon, but also the extent to which advanced, complex, and powerful
societies are susceptible to collapse. The fall of the Roman Empire, and
the equally (if not more) advanced Han, Mauryan, and Gupta Empires,
as well as so many advanced Mesopotamian Empires, are all testimony
to the fact that advanced, sophisticated, complex, and creative civilizations can be both fragile and impermanent.
A large number of explanations have been proposed for each specific
case of collapse, including one or more of the following: volcanoes,
earthquakes, droughts, floods, changes in the courses of rivers, soil
91
degradation (erosion, exhaustion, salinization, etc.), deforestation, climate change, tribal migrations, foreign invasions, changes in technology
(such as the introduction of ironworking), changes in the methods or
weapons of warfare (such as the introduction of horse cavalry, armored
infantry, or long swords), changes in trade patterns, depletion of particular mineral resources (e.g., silver mines), cultural decline and social
decadence, popular uprisings, and civil wars. However, these explanations are specific to each particular case of collapse rather than general.
Moreover, even for the specific case where the explanation applies, the
society in question usually had already experienced the phenomenon
identified as the cause without collapsing. For example, the Minoan society had repeatedly experienced earthquakes that destroyed palaces,
and they simply rebuilt them more splendidly than before. Indeed,
many societies experience droughts, floods, volcanoes, soil erosion,
and deforestation with no major social disruption (Tainter, 1988).
The same applies to migrations, invasions, and civil wars. The
Roman, Han, Assyrian, and Mauryan Empires were, for centuries, completely militarily hegemonic, successfully defeating the neighboring
“barbarian” peoples who eventually did overrun them. So external
military pressure alone hardly constitutes an explanation for their
collapses. With both natural disasters and external threats, identifying
a specific cause compels one to ask, “yes, but why did this particular
instance of this factor produce the collapse?” Other processes must be
involved, and, in fact, the political, economic, ecological, and technological conditions under which civilizations have collapsed have varied
widely. Individual collapses may have involved an array of specific factors, with particular triggers, but a general explanation remains elusive.
Individual explanations may seem appropriate in their particular case,
but the very universal nature of the phenomenon implies a mechanism
that is not specific to a particular time period of human history, nor a
particular culture, technology, or natural disaster (Tainter, 1988;
Turchin, 2003; Yoffee and Cowgill, 1988).
In this paper we attempt to model collapse mathematically in a more
general way. We propose a simple model, not intended to describe actual individual cases, but rather to provide a general framework that
allows carrying out “thought experiments” for the phenomenon of
collapse and to test changes that would avoid it. This model (called
HANDY, for Human and Nature DYnamics) advances beyond existing
biological dynamic population models by simultaneously modeling
two separate important features which seem to appear across so
many societies that have collapsed: (1) the stretching of resources due
to the strain placed on the ecological carrying capacity (Abel, 1980;
Catton, 1980; Kammen, 1994; Ladurie, 1987; Ponting, 1991; Postan,
1966; Redman, 1999; Redman et al., 2004; Wood, 1998; Wright,
2004), and (2) the economic stratification of society into Elites and
Masses (or “Commoners”) (Brenner, 1985; Parsons, 1991; Turchin,
2005, 2006; Turchin and Nefedov, 2009; Diamond, 2005; Goldstone,
1991; Ibn Khaldun, 1958). In many of these historical cases, we have direct evidence of Ecological Strain and Economic Stratification playing a
central role in the character or in the process of the collapse (Culbert,
1973; Diamond, 2005; Goldstone, 1991; Lentz, 2000; Mitchell, 1990).
For these empirical reasons, and the theoretical ones explained in
Section 3, our model incorporates both of these two features. Although
similar to the Brander and Taylor (1998) model (hereafter referred to as
“BT”) in that HANDY is based on the classical predator–prey model, the
inclusion of two societal classes introduces a much richer set of dynamical solutions, including cycles of societal and ecological collapse, as well
as the possibility of smoothly reaching equilibrium (the ecological carrying capacity). We use Carrying Capacity in its biological definition:
the population level that the resources of a particular environment
can sustain over the long term (Catton, 1980; Cohen, 1995; Daly and
Farley, 2003). In this paper, we call these environment resources
“Nature”.
The paper is organized as follows: Section 2 gives a brief review of
the predator–prey model; Section 3 includes the mathematical description of HANDY; Section 4 covers a theoretical analysis of the model
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S. Motesharrei et al. / Ecological Economics 101 (2014) 90–102
equilibrium and possible solutions; Section 5 presents examples of scenarios within three distinct types of societies; Section 6 gives an overall
discussion of the scenarios from Section 5; and Section 7 offers a short
summary of the paper and a discussion of future work.
2. Predator–Prey Model
The predator–prey model, the original inspiration behind HANDY,
was derived independently by two mathematicians, Alfred Lotka and
Vitto Volterra, in the early 20th century (Lotka, 1925; Volterra, 1926).
This model describes the dynamics of competition between two species,
say, wolves and rabbits. The governing system of equations is
ẋ ¼ ðayÞx−bx
:
ẏ ¼ cy−ðdxÞy
ð1Þ
In the above system, x represents the predator (wolf) population; y
represents the prey (rabbit) population; a determines the predator's
birth rate, i.e., the faster growth of wolf population due to availability
of rabbits; b is the predator's death rate; c is the prey's birth rate; d determines the predation rate, i.e., the rate at which rabbits are hunted by
wolves.
Rather than reaching a stable equilibrium, the predator and prey
populations show periodic, out-of-phase variations about the equilibrium values
xe
ye
¼
¼
c=d
:
b=a
ð2Þ
Note consistency of the units on the left and right hand sides of
Eqs. (1) and (2). A typical solution of the predator–prey system can be
seen in Fig. 1.
3. HANDY
As indicated above, Human And Nature DYnamics (HANDY) was
originally built based on the predator–prey model. We can think of
the human population as the “predator”, while nature (the natural
resources of the surrounding environment) can be taken as the “prey”,
depleted by humans. In animal models, carrying capacity is an upper
ceiling on long-term population. When the population surpasses the
carrying capacity, mechanisms such as starvation or migration bring
the population back down. However, in the context of human societies,
400 wolves
2,000 rabbits
Predator (wolves) and Prey (rabbits) Populations
Predator
200 wolves
1,000 rabbits
0 wolves
0 rabbits
Prey
Time (year)
Fig. 1. A typical solution of the predator–prey system obtained by running the system with the following parameter values and initial conditions: a = 3.0 × 10− 5 (rabbits·years)− 1; b = 2.0 × 10− 2 years− 1,
c = 3.0 × 10− 2 years− 1, d = 2.0 × 10− 4 (wolves·years)− 1; x(0) =
1.0 × 10+ 2 wolves; and y(0) = 1.0 × 10+ 3 rabbits. Predator population is measured in units of wolves, Prey population is measured
in units of rabbits, and Time is measured in units of years.
the population does not necessarily begin to decline upon passing the
threshold of carrying capacity, because, unlike animals, humans can
accumulate large surpluses (i.e., wealth) and then draw down those resources when production can no longer meet the needs of consumption.
This introduces a different kind of delay that allows for much more complex dynamics, fundamentally altering the behavior and output of the
model. Thus, our model adds the element of accumulated surplus not
required in animal models, but which we feel is necessary for human
models. We call this accumulated surplus “wealth”.
Empirically, however, this accumulated surplus is not evenly distributed throughout society, but rather has been controlled by an elite. The
mass of the population, while producing the wealth, is only allocated a
small portion of it by elites, usually at or just above subsistence levels.
Based on this, and on the historical cases discussed in the introduction,
we separated the population into “Elites” and “Commoners”, and introduced a variable for accumulated wealth. For an analysis of this twoclass structure of modern society, see Drăgulescu and Yakovenko
(2001) and Banerjee and Yakovenko (2010). This adds a different dimension of predation whereby Elites “prey” on the production of wealth
by Commoners. As a result, HANDY consists of four prediction equations: two for the two classes of population, Elites and Commoners,
denoted by xE and xC, respectively; one for the natural resources or
Nature, y; and one for the accumulated Wealth, w, referred to hereafter
as “Wealth”. This minimal set of four equations seems to capture essential features of the human–nature interaction and is capable of producing major potential scenarios of collapse or transition to steady state.
A similar model of population and renewable resource dynamics
based on the predator–prey model was developed in the pioneering
work of Brander and Taylor (1998) demonstrating that reasonable
parameter values can produce cyclical “feast and famine” patterns of
population and resources. Their model showed that a system with a
slow-growing resource base will exhibit overshooting and collapse,
whereas a more rapidly growing resource base will produce an adjustment of population and resources toward equilibrium values. They
then applied this model to the historical case of Easter Island, finding
that the model provides a plausible explanation of the population
dynamics known about Easter Island from the archeological and scientific record. They thus argue that the Polynesian cases where population
did collapse were due to smaller maximum resource bases (which they
call “carrying capacity”) that grew more slowly, whereas those cases
which did not experience such a collapse were due to having a larger resource base (i.e., a larger carrying capacity). They then speculate that
their model might be consistent with other historical cases of collapse,
such as the ancient Mesopotamian and Maya civilizations or modern
Rwanda.
However, the BT approach only models Population and Nature and
does not include a central component of these historical cases: economic stratification and the accumulation of wealth. Thus, despite clear evidence for a stratified class structure in Easter Island's history prior to the
collapse (as well as for Mesopotamia, the ancient Maya, and modern
Rwanda), the BT model does not include class stratification as a factor.
In their model, society produces and consumes as a single homogeneous
unit. We feel that a historically realistic modeling of the evolution of
human–nature dynamics in these stratified complex societies cannot
be achieved without including this class stratification in the model.
Brander and Taylor recognize that their model is simple, and that application to more complex scenarios may require further development of
the structure of the model. We have found that including economic
stratification, in the form of the introduction of Elites and Commoners,
as well as accumulated Wealth, results in a much richer variety of solutions, which may have a wider application across different types of societies. HANDY's structure also allows for “irreversible” collapses, without
the need to introduce an explicit critical depensation mechanism into
the model as other models need to do. Thus while the Brander–Taylor
model has only two equations, HANDY has four equations to predict
the evolution of the rich and poor populations (Elites and Commoners),
S. Motesharrei et al. / Ecological Economics 101 (2014) 90–102
Nature, and accumulated Wealth (we examine other differences in
Section 6.4 of the paper) The HANDY equations are given by:
8
ẋ ¼ βC xC −α C xC
>
>
< C
ẋ E ¼ βE xE −α E xE
>
>
: ẏ¼ γyðλ−yÞ−δxC y
ẇ¼ δxC y−C C −C E :
ð3Þ
It is to be noted that αC, αE, CC, and CE are all functions of w, xC, and xE.
See Eqs. (4) and (6) and Fig. 2a and b.
3.1. Model Description
The total population is divided between the two variables, xC and xE,
representing the population of commoners and of elites. The population
grows through a birth rate β and decreases through a death rate α. β is
assumed to be constant for both Elites and Commoners but α depends
on Wealth as explained below.
In reality, natural resources exist in three forms: nonrenewable
stocks (fossil fuels, mineral deposits, etc.), regenerating stocks (forests,
soils, animal herds, wild fish stocks, game animals, aquifers, etc.), and
renewable flows (wind, solar radiation, precipitation, rivers, etc.). Future generations of the model will disaggregate these forms. We have
adopted a single formulation intended to represent an amalgamation
of the three forms, allowing for a clear understanding of the role that
natural resources play in collapse or sustainability of human societies.
Thus, the equation for Nature includes a regeneration term,
γy(λ − y), and a depletion term, − δxCy. The regeneration term has
been written in the form of a logistic equation, with a regeneration factor, γ, exponential regrowth for low values of y, and saturation when y
approaches λ, Nature's capacity — maximum size of Nature in absence
of depletion. As a result, the maximum rate of regeneration takes
place when y = λ / 2. Production is understood according to the standard Ecological Economics formulations as involving both inputs from,
and outputs to, Nature (i.e., depletion of natural sources and pollution
of natural sinks) (Daly, 1996; Daly and Farley, 2003). This first generation of HANDY models the depletion side of the equation as if it includes
the reduction in Nature due to pollution.
The depletion term includes a rate of depletion per worker, δ, and
is proportional to both Nature and the number of workers. However,
the economic activity of Elites is modeled to represent executive,
management, and supervisory functions, but not engagement in the direct extraction of resources, which is done by Commoners. Thus, only
Commoners produce.
a) Consumption rates in HANDY
93
It is frequently claimed that technological change can reduce resource
depletion and therefore increase carrying capacity. However, the effects
of technological change on resource use are not unidirectional. Technological change can raise the efficiency of resource use, but it also tends to
raise both per capita resource consumption and the scale of resource extraction, so that, absent policy effects, the increases in consumption
often compensate for the increased efficiency of resource use. These
are associated with the phenomena referred to as the Jevons Paradox,
and the “Rebound Effect” (Greening et al., 2000; Polimeni et al., 2008;
Ruth, 2009). For example, an increase in vehicle fuel efficiency tends
to enable increased per capita vehicle miles driven, heavier cars, and
higher average speeds, which then negate the gains from the increased
fuel-efficiency. In addition, technological advances can enable greater
resource extraction and throughput, which then appears as increases
in the productivity of other factors of production. As Daly points out,
much of the increase in productivity in both agriculture and industry
in the last two centuries has actually come from increased (rather
than decreased) resource throughput (Daly, 1991). A decline in the
price of a resource is usually thought to reflect an increase in the abundance of that resource, but in fact, it often reflects that the resource is
simply being extracted more rapidly. Rather than extend carrying capacity, this reduces it. Over the long-term, per capita resource-use has
tended to rise over time despite dramatic technological advances in resource efficiency. Thus, the sign and magnitude of the effect of technological change on resource use varies and the overall effect is difficult
to predict. Therefore, in this generation of HANDY, we assume that the
effects of these trends cancel each other out. The model will be developed further to allow the rates of these technology-induced trends to
be adjusted in either direction.
Finally, there is an equation for accumulated Wealth, which increases
with production, δxCy, and decreases with the consumption of the Elites
and the Commoners, CC and CE, respectively. The consumption of the
Commoners (as long as there is enough wealth to pay them) is sxC, a
subsistence salary per capita, s, multiplied by the working population.
The Elites pay themselves a salary κ times larger, so that the consumption of the Elites is κsxE. However, when the wealth becomes too small
to pay for this consumption, i.e., when w b wth, the payment is reduced
and eventually stopped, and famine takes place, with a much higher rate
of death. κ is meant to represent here the factors that determine the division of the output of the total production of society between elites and
masses, such as the balance of class power between elites and masses,
and the capacity of each group to organize and pursue their economic
interests. We recognize the inherent limitations, in this initial generation
of our model, of holding that balance (κ) constant in each scenario, but
we expect to develop κ further in later generations of HANDY so that it
can be endogenously determined by other factors in the model.
b) Death rates in HANDY
C
Fig. 2. Per capita Consumption rates and Death rates for Elites and Commoners as a function of Wealth. Famine starts when sx
≤1. Therefore, Comw
w
1
moners start experiencing famine when wth ≤1, while Elites do not experience famine until wth ≤ κ . This delay is due to Elites' unequal access to
Wealth.
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S. Motesharrei et al. / Ecological Economics 101 (2014) 90–102
CC and CE, the consumption rates for the Commoner and the Elite respectively, are given by the following equations:
8
>
>
< CC
>
>
: CE
w
sxC
min 1;
wth
:
w
¼ min 1;
κsxE
wth
4.1. Equilibrium when xE = 0 (No Elites): Egalitarian Society
¼
ð4Þ
Wealth threshold, wth, is a threshold value for wealth below which
famine starts. It depends on the “minimum required consumption per
capita”, ρ:
wth ¼ ρxC þ κρxE :
¼
>
>
: αE
¼
C
α m þ max 0; 1− C ðα M −α m Þ
sxC :
C
α m þ max 0; 1− E ðα M −α m Þ
sxE
ð6Þ
The death rates vary between a normal (healthy) value, αm, observed when there is enough food for subsistence, and a maximum
(famine) value, αM that prevails when the accumulated wealth has
been used up and the population starves. There are a variety of mechanisms which can reduce population when it exceeds carrying capacity,
including everything from emigration, increased disease susceptibility,
and outright starvation to breakdowns in social order and increased social violence, such as banditry, riots, rebellions, revolutions, and wars.
These mechanisms are described in detail in Turchin (2003) but the
net effect of all of them is a reduction in population, and that is what
the dynamics of our model is meant to represent when we say “population decline” or “famine”. Note also that an increase in the death rates
(α) is equivalent to an equal decrease in the birth rates (β). The death
rates αC and αE can be expressed in terms of ww , a graphical representation of which is given Fig. 2b.
th
There are three dimensions for quantities in HANDY:
y
>
>
: e
we
γ
s
λ−η
δ
δ
s
:
¼ η
δ
¼ ηρxC;e
¼
ð8Þ
We define χ, the Carrying Capacity for the population, to be equal to
xC,e in Eq. (8), i.e., the equilibrium value of the population in the absence
of Elites:
χ¼
γ
s
λ−η :
δ
δ
ð9Þ
Carrying Capacity can be maximized if Nature's regeneration rate is
maximal, i.e., if ye ¼ λ2. This requires δ to be set equal to a value δ* that
can result in a steady state with the maximum (sustainable) Population,
which in this paper we call the “optimal” value of δ. From the second
equation in Eq. (8), it can be seen that δ* is given by:
δ ¼
2ηs
:
λ
ð10Þ
The Maximum Carrying Capacity, χM, is thus given by:
χM ¼
γλ γ λ 2
¼
:
δ 2 ηs 2
ð11Þ
4.2. Equilibrium when xE ≥ 0 and κ = 1 (No Inequality): Equitable Society
If we set κ ≡ 1 and βE ≡ βC ≡ β, we can reach an equilibrium state for
which xE ≥ 0. This case models an equitable society of “Workers” and
“Non-Workers”. We need a dimensionless free parameter φ that sets
the initial ratio of the Non-Workers to Workers:
xE ð0Þ
:
xC ð0Þ
ð12Þ
The equilibrium values of the system can then be expressed as
follows:
1. Population (either Commoner or Elite), in units of people.
2. Nature/Wealth, in units of “eco-Dollars”.
3. Time, in units of years.
The structure of the model requires Nature and Wealth to be measured with the same units, therefore we created the unit eco-dollar.
Other parameters and functions in the model carry units that are compatible with the abovementioned dimensions following Eq. (3). For example, Carrying Capacity, χ, and the Maximum Carrying Capacity, χM,
defined in Section 4.1, are both expressed in units of people.
4. Equilibrium Values and Carrying Capacity
8
>
xC;e
>
>
>
<x
E;e
>
> ye
>
>
:
we
ð7Þ
γ
s
λ−η ð1 þ φÞ
¼
δ
δ
¼ φxC;e
:
s
¼ η ð1 þ φÞ
δ
¼ ηρð1 þ φÞxC;e
ð13Þ
The total population xe = xC,e + xE,e can still be maximized by choosing δ appropriately:
δ ¼
We can use the model to find a sustainable equilibrium and maximum carrying capacity in different types of societies. In order for population to reach an equilibrium, we must have αm ≤ βE ≤ βC ≤ αM. We
define a dimensionless parameter, η:
α M −βC
:
α M −α m
8
>
>
< xC;e
φ¼
3.2. A Note on Units and Dimensions
η¼
Assuming xE ≡ 0, we can find the equilibrium values of the system
(subscript “e” denotes the equilibrium values):
ð5Þ
Even when Commoners start experiencing famine, i.e., when w ≤
wth, the Elites continue consuming unequally as indicated by the factor
κ in the second term on the right hand side of Eq. (5). A graphical representation of the consumption rates are given in Fig. 2a.
The death rates for the Commoner and the Elite, αC and αE, are functions of consumption rates:
8
>
>
< αC
Since we assume αm ≤ βC ≤ αM, η will always be bounded by 0 ≤
η ≤ 1.
2ηs
ð1 þ φÞ:
λ
ð14Þ
This δ** is larger than the optimal depletion factor given by Eq. (10).
The difference arises because Workers have to produce more than they
need just for themselves in order to support Non-Workers. For this
choice of δ, total population is given by:
xe;M ¼ ð1 þ φÞ
γ λ γ λ 2
¼
:
δ 2 ηs 2
ð15Þ
S. Motesharrei et al. / Ecological Economics 101 (2014) 90–102
As can be seen from Eq. (15), maximum total population in equilibrium is independent of φ and conforms to the maximum carrying capacity given above by Eq. (11).
4.3. Equilibrium when xE ≥ 0 and κ N 1: Unequal Society
It is possible to attain equilibrium in an unequal society if we can satisfy the following condition:
α M −βE
α −βC
¼ M
¼ η:
κ ðα M −α m Þ α M −α m
ð16Þ
(The general condition αm ≤ βE ≤ βC ≤ αM must hold in all cases for
an equilibrium to be feasible.)
The equilibrium values in this general case can be expressed as
follows:
8
γ
s
>
>
xC;e ¼
λ−η ð1 þ κψÞ
>
>
δ
δ
>
< x ¼ ψx
E;e
C;e
:
s
>
> ye ¼
η ð1 þ κψÞ
>
>
δ
>
: w ¼ ηρð1 þ κψÞx
e
C;e
ð17Þ
The free parameter, ψ, is the equilibrium ratio xE,e/xC,e, apparent from
the second equation in Eq. (17). As opposed to φ, ψ cannot be easily related to the initial conditions; rather, it can be determined from the result of a simulation.
Again, the total population xe = xC,e + xE,e can be maximized by
choosing δ appropriately:
δ ¼
2ηs
ð1 þ κψÞ:
λ
ð18Þ
This required depletion rate δ*** can be even larger than the optimal
δ given by Eq. (14) depending upon the values of κ and ψ. In the
presence of inequality, the maximum total population is no longer independent of κ and ψ and is smaller than the maximum carrying capacity
given by Eqs. (11) and (15):
xe;M ¼ ð1 þ ψÞ
γ λ γ λ 2 1þψ
¼
:
δ 2 ηs 2
1 þ κψ
ð19Þ
5. Scenarios
We discuss three sets of scenarios:
1. Egalitarian society (No-Elites): Scenarios in which xE = 0.
2. Equitable society (with Workers and Non-Workers): Scenarios in
which xE ≥ 0 but κ ≡ 1.
3. Unequal society (with Elites and Commoners): Scenarios in which
xE ≥ 0 and κ N 1.
For all of these scenarios, we start the model with the typical parameter values and initial conditions given in Table 1, unless otherwise stated. As indicated above, the values of κ and xE(0) determine the type of
the society. Within each type of society, we obtain different scenarios
by varying the depletion factor, δ.
In this section, we will show that HANDY is capable of modeling
three distinct types of societies by changing κ and xE(0). A sustainable
equilibrium can be found for each society by controlling δ. An appropriate choice of δ can make this equilibrium optimal, i.e., with maximum
total population. Increasing δ above its optimal value makes the approach toward equilibrium oscillatory. Such an equilibrium is suboptimal, and the Carrying Capacity is below its maximum value, χM. It is
also possible to reach a suboptimal equilibrium (a less than maximum,
but sustainable population) by making δ lower than its optimal value.
95
Table 1
Description of parameters and state variables used in HANDY. κ, δ, and xE are varied to
study various scenarios in three different types of societies. xE = 0 defines an Egalitarian
society with no Elites. κ = 1 defines an Equitable society with Workers and NonWorkers, represented by xC and xE in this case, respectively. xE ≥ 0 and κ N 1 define an unequal society with Elites and Commoners (xE and xC). As a reference, all other variables and
functions in HANDY are also listed above. Subscript e denotes equilibrium value everywhere in this paper.
Parameter symbol
Parameter name
Typical value(s)
αm
αM
βC
βE
s
ρ
γ
λ
κ
δ
Normal (minimum) death rate
Famine (maximum) death rate
Commoner birth rate
Elite birth rate
Subsistence salary per capita
Threshold wealth per capita
Regeneration rate of nature
Nature carrying capacity
Inequality factor
Depletion (production) factor
1.0 × 10−2
7.0 × 10−2
3.0 × 10−2
3.0 × 10−2
5.0 × 10−4
5.0 × 10−3
1.0 × 10−2
1.0 × 10+2
1, 10, 100
None
Variable symbol
Variable name
Typical initial value(s)
xC
xE
y
w
Commoner population
Elite population
Nature
Accumulated wealth
1.0 × 10+2
0, 1, 25
λ
0
(a). List of parameters in HANDY. κ and δ take different values for different scenarios.
(b). List of state variables in HANDY. xE(0) takes different values for different scenarios.
However, in the latter case, the approach toward equilibrium would
be a soft landing rather than oscillatory. When δ is increased even
further, the society goes into cycles of prosperity and collapse. Increasing δ beyond a certain point will result in an irreversible Type-N (full)
collapse, examples of which are presented in Sections 5.1.4, 5.2.4,
and 5.3.2. We give a full categorization of collapses in the next two
paragraphs.
Running the model in different scenarios produces two kinds of collapses, either due to scarcity of labor (following an inequality-induced
famine) or due to scarcity of Nature (depletion of natural resources).
We categorize the former case as a Type-L (Disappearance of Labor)
Collapse and the latter as a Type-N collapse (Exhaustion of Nature). In
a Type-L collapse, growth of the Elite Population strains availability of
resources for the Commoners. This causes decline of the Commoner
Population (which does the labor), and consequently, decline of Wealth.
Finally, Elite Population plummets since its source of subsistence,
i.e., Wealth, has vanished. See Fig. 6a for an example of a Type-L collapse. This could represent a historical case such as the disappearance
of the Mayan civilization in the Yucatan. Note that this type of collapse
can only happen in an unequal society, because the major cause behind
it is inequality.
Table 2
As a reference, all other variables and functions in HANDY are listed in this table. Subscript
e denotes equilibrium value everywhere in this paper.
Variable
symbol
Variable name
Defining equation
wth
ω
CC
CE
αC
αE
η
χ
δ*
χM
φ
δ**
ψ
δ***
Threshold wealth
Normalized wealth
Commoner consumption
Elite consumption
Commoner death rate
Elite death rate
η
Carrying Capacity (CC)
Egalitarian optimal δ
Maximum Carrying Capacity (Max CC)
Ratio of non-workers to workers (Equitable)
Equitable optimal δ
Elite to commoner equilibrium ratio (Unequal)
Unequal optimal δ
Eq. (5)
w / wth
Eq. (4) (Fig. 2a)
Eq. (4) (Fig. 2a)
Eq. (6) (Fig. 2b)
Eq. (6) (Fig. 2b)
Eq. (7)
Eq. (9)
Eq. (10)
Eq. (11)
Eq. (12)
Eq. (14)
xE,e / xC,e
Eq. (18)
96
S. Motesharrei et al. / Ecological Economics 101 (2014) 90–102
1
1
4
Egalitarian Society: Soft Landing to Optimal Equilibrium
Carrying
Capacity
2
1
20
Commoners
Nature
Wealth
Nature
0.5
0.5
2
1
0.5
10
0
0
0
0
0
0
Time (Year)
a) Soft landing to the optimal equilibrium when Elite
population (marked in red) equals zero. Final population
reaches the carrying capacity, which is at its maximum
value,
, in this scenario.
2
1
20
Egalitarian Society: Cycles of Prosperity and Reversible,
Type-N Collapses
Nature
Commoners
Carrying
Capacity
Commoners
Wealth
Time (Year)
b) Oscillatory approach to equilibrium when Elite population (marked in red) equals zero. Final population converges to the carrying capacity, which is lower than its
maximum value,
, in this scenario.
2
1
20
Egalitarian Society: Irreversible, Type-N (Full) Collapse
Nature
Wealth
Wealth
1
0.5
10
1
0.5
10
0
0
0
Egalitarian Society: Oscillatory Approach to Equilibrium
Carrying
Capacity
Commoners
Carrying Capacity
0
0
0
Time (Year)
Time (Year)
c) Cycles of prosperity, overshoot, (reversible Type-N)
collapse, and revival when Elite population (marked in
red) equals zero.
d) Irreversible Type-N collapse (full collapse) when Elite
population (marked in red) equals zero. All the state variables collapse to zero in this scenario due to overdepletion.
Fig. 3. Experiment results for the Egalitarian society.
A Type-N collapse, on the other hand, starts with an exhaustion
of Nature, followed by a decline of Wealth that in turn, causes a fall
of the Commoners and then the Elites. Depending on the depletion
rate, Type-N collapses can be “reversible” or “irreversible”. After a
reversible collapse, regrowth of nature can trigger another cycle of
prosperity, examples of which can be seen in Figs. 3c and 4c. This
could represent historical cases such as the Greek and Roman
collapses.
When depletion is pushed beyond a certain limit, Nature fully
collapses and the whole system completely collapses after that.
This is why we call an irreversible Type-N collapse a “full” collapse.
Examples of such collapses can be seen in Figs. 3d, 4d, and 6b. This
could represent a historical case such as the exhaustion of Nature
on Easter Island. Type-N collapses can arise because of excessive depletion only (Figs. 3d and 4d), or both excessive depletion and inequality (Fig. 6b).
It is important to understand the inter-relation of the depletion factor, δ, and the Carrying Capacity, χ. The further δ is taken away from its
optimal value, the further χ moves down from its maximum value, χM.
An equilibrium can be reached if and only if χ is not too far away from
χM, which means δ cannot be too far away from its optimal value,
given by Eqs. (10), (14), and (18) in the three types of societies under
consideration. Note that in all of the scenario outputs presented below
(for the three types of societies under consideration), Carrying Capacity
(χ) and the Maximum Carrying Capacity (χM) are calculated from their
defining Eqs. (9) and (11), respectively.
Important note about the units of the vertical axis of all the subsequent graphs: Populations, xC and xE, and the Carrying Capacity, χ, are
all normalized to the Maximum Carrying Capacity, χM. Nature and
Wealth are both shown in units of Nature's capacity, λ. The top scale
of the vertical axis of the graph pertains to Population(s) and Carrying
Capacity; the middle scale pertains to Nature, which (normally) stays
bounded by 1λ; and the bottom scale is for Wealth.
Note: All the simulations below use the Euler integration method
with a time-step of 1 year and single precision.
5.1. Egalitarian Society (No-Elites): xE = 0
In the four following scenarios, κ does not play any role since we
set xE ≡ 0. We start the depletion rate from δ = δ∗, the optimal equilibrium value that maximizes the Carrying Capacity, and increase it
slowly to get additional scenarios. The horizontal red line in the
graphs for the four scenarios of this section represents the zero population of Elites.
5.1.1. Egalitarian Society: Soft Landing to Equilibrium
For the scenario in Fig. 3a, δ = δ∗ = 6.67 × 10−6. Therefore, the carrying capacity, χ, is at its maximum level, χM. Notice that Nature also
settles to ye = λ / 2, which is the value that results in the maximum regeneration rate. This maximal regeneration can in turn support a maximum sustainable depletion and population.
If we set δ b δ*, we still see a soft landing to the carrying capacity, χ.
However, χ would be at a lower level than χM because a lower-thanoptimal δ does not correspond to the maximum regeneration of nature,
which is a necessity if we want to have the maximum sustainable
S. Motesharrei et al. / Ecological Economics 101 (2014) 90–102
population. The advantage of a lower-than-optimal δ is a higher equilibrium level (compared to λ / 2) for Nature.
Choosing a depletion rate, δ, that is too small to produce enough to
feed the population would result in a collapse, and thus make any equilibrium impossible even though Nature stays at its maximum capacity.
Of course, this would not occur in the real world as the urge for survival
guarantees humans extract their basic needs from nature.
5.1.2. Egalitarian Society: Oscillatory Approach to Equilibrium
For the scenario in Fig. 3b, δ is increased to δ = 2.5δ∗ = 1.67 × 10−5.
As can be seen from Fig. 3b, the carrying capacity, χ, is lower than its
maximum value, χM. Population initially overshoots the carrying capacity, then oscillates, and eventually converges to it since the amount of
overshoot is not too large, just about the order of χ. Note that at the
time the (total) population overshoots the Carrying Capacity, the
Wealth also reaches a maximum and starts to decline.
5.1.3. Egalitarian Society: Cycles of Prosperity, Overshoot, Collapse, and
Revival
For the scenario in Fig. 3c, δ is increased to δ = 4δ∗ = 2.67 × 10−5. As
can be seen, Population, Nature and Wealth all collapse to a very small
value. However, after depletion becomes small due to very low number
of workers, Nature gets a chance to grow back close to its capacity, λ.
The regrowth of Nature kicks off another cycle of prosperity which
ends with another collapse. Simulation results show that these cycles,
ending in Type-N collapses (i.e., those that start due to scarcity of
Nature), repeat themselves indefinitely. Therefore, such cycles represent “reversible” Type-N collapses. This reversibility is possible as long
as δ stays within a “safe” neighborhood of δ*.
1
1
4
Equitable Society: Soft Landing to Optimal Equilibrium
Carrying
Capacity
Workers
Wealth
0.5
0.5
2
Nature
Non-Workers
0
0
0
5.1.4. Egalitarian Society: Irreversible Type-N Collapse (Full Collapse)
For the scenario in Fig. 3d, δ is increased further to δ = 5.5δ∗ =
3.67E − 5. The overshoot is so large that it forces Population, Nature
and Wealth into a full collapse, after which there is no recovery. This
is a generic type of collapse that can happen for any type of society
due to over-depletion. See Sections 5.2.4 and 5.3.2 for examples of irreversible Type-N collapses in equitable and unequal societies, respectively. We include further discussion of these two types of collapses in
Section 6.
We observe that the accumulated Wealth delays a decline of the
population even after Nature has declined well below its capacity, λ.
Therefore, population keeps growing and depleting Nature until Nature
is fully exhausted. At that instance, i.e., when y = 0, Wealth cannot grow
any further; indeed, it starts plummeting, causing a sharp fall of the
population level, and eventually its full, irreversible collapse.
5.2. Equitable Society (with Workers and Non-Workers): κ = 1
We take the parameter values and the initial conditions to be the
same as in Table 1, except that this time we set xE(0) = 25 (φ = 0.25)
and κ = 1. We start with the optimal depletion per capita δ = δ**,
which will sustain the maximum population (see Eq. (14)), and will
gradually increase it in order to get the additional scenarios in this subsection. Notice that in these cases, xC describes the Working Population,
while xE describes the Non-Working Population. Everybody consumes
at the same level, since we set κ = 1, i.e., we assume there is no inequality in consumption level for Workers and Non-Workers.
5.2.1. Equitable Society: Soft Landing to Optimal Equilibrium
For the scenario in Fig. 4a, δ = δ∗∗ = 8.33 × 10−6. Notice that this
is larger than the optimal value in the absence of Non-Workers δ∗ =
1
1
20
0.5
0.5
10
0
0
0
Equitable Society: Cycles of Prosperity and Reversible,
Type-N Collapses
Workers
1
1
20
Nature
0.5
0.5
10
Carrying
Capacity
0
0
0
Wealth
Equitable Society: Oscillatory Approach to Equilibrium
Nature
Carrying
Capacity
Wealth
Non-Workers
Equitable Society: Irreversible, Type-N (Full) Collapse
Nature
0.5
0.5
10
Non-Workers
Workers
Time (Year)
b) Oscillatory approach to equilibrium in the presence
of both Workers and Non-Workers is possible when the
overshoot is not too large.
Time (Year)
a) Equilibrium in the presence of both Workers and NonWorkers can be attained with slow growth and equitable
salaries.
1
1
20
97
0
0
0
Workers
Carrying Capacity
Wealth
Non-Workers
Time (Year)
Time (Year)
c) Cycles of prosperity, overshoot, (reversible Type-N)
collapse, and revival in the presence of Workers and NonWorkers.
d) Irreversible Type-N collapse (full collapse) happens
after a period of very fast growth.
Fig. 4. Experiment results for the Equitable society.
98
S. Motesharrei et al. / Ecological Economics 101 (2014) 90–102
1
1
4
Equitable Society: Preventing a Full Collapse
Non-Workers
Wealth
0.5
0.5
2
Nature
Carrying Capacity
0
0
0
Workers
Time (Year)
Fig. 5. The full collapse that happened in the previous scenario, Fig. 4d,
can be prevented by reducing the average depletion per capita. This
can be achieved by either increasing the ratio of the Non-Working to
Working population (high δ, high φ) or decreasing the average workload per worker, i.e., decreasing total work hours per week (low δ, low φ).
6.67 × 10−6 even though all the other parameters are identical to those
in Section 5.1.1. This difference arises because xE ≠ 0, which in turn
forces the Workers to produce extra in order to support the NonWorkers. Now, χ b χM because δ = δ∗∗ ≠ δ∗. However, by setting δ =
δ**, the optimal value of δ in the presence of Non-Workers, the total population, xC + xE still reaches the maximum Carrying Capacity, χM, the
same as in Section 5.1. See Eq. (15) and Section 4.2 for a mathematical
description.
6
1
4
Unequal Society: Type-L Collapse (Scarcity of Labor)
Similar comments as in Section 5.1.1 apply here when we choose a
lower-than-optimal δ.
5.2.2. Equitable Society: Oscillatory Approach to Equilibrium
For the scenario in Fig. 4b, δ = 2.64δ∗∗ = 2.20 × 10−5. The total population is equal to the actual Carrying Capacity (smaller than the maximum Carrying Capacity).
5.2.3. Equitable Society: Cycles of Prosperity, Overshoot, Collapse,
and Revival
For the scenario in Fig. 4c, δ = 3.46δ∗∗ = 3.00 × 10−5. The result is
analogous to Fig. 3c which corresponds to Section 5.1.3. As before, the
time at which the total population overshoots the actual Carrying Capacity is indicated by the fact that Wealth starts to decrease. After
each cycle of prosperity, there is a partial, reversible Type-N collapse.
5.2.4. Equitable Society: Full Collapse
For the scenario in Fig. 4d, δ = 5δ∗∗ = 4.33 × 10−5. Once again, we
can see how an irreversible Type-N (full) collapse of Population, Nature,
and Wealth can occur due to over-depletion of natural resources as a result of high depletion per capita.
5.2.5. Equitable Society: Preventing a Full Collapse by Decreasing Average
Depletion per Capita
The case in Fig. 5 is similar to the previous case (see Section 5.2.4 and
Fig. 4d), except that we raised the ratio of Non-Workers to Workers, φ,
from 0.25 to 6. This corresponds to changing xE(0) from 25 to 600,
2
1
40
Unequal Society: Irreversible, Type-N (Full) Collapse
Nature
3
0.5
2
0
0
0
Wealth
Nature
(Equivalent)
Elites
Carrying
Capacity
0
0
0
Commoners
Wealth
1
0.5
20
(Equivalent)
Elites
Carrying
Capacity
Time (Year)
Time (Year)
1
1
4
0.5
0.5
2
0
0
0
Commoners
a) Population collapse following an apparent
b) A fast full collapse due to both over-
equilibrium due to a small initial Elite population when
100. This scenario also shows
a different route to a collapse, in which, although Nature eventually recovers, population
does not.
depletion and inequality
Unequal Society: Soft Landing to Optimal Equilibrium
(Equivalent) Elites
Unequal Society: Oscillatory Approach to Equilibrium
Nature
Carrying Capacity
Commoners
Wealth
Nature
(Equivalent)
Carrying Capacity
Commoners
Elites
Wealth
Time (Year)
Time (Year)
c) With moderate in equality
, it is possible to attain an optimal equilibrium by controlling the birth rates.
d) With
, it is still possible to oscillate
and converge to an equilibrium
Fig. 6. Experiment results for the Unequal society.
S. Motesharrei et al. / Ecological Economics 101 (2014) 90–102
while keeping xC(0) = 100. By increasing the ratio of non-workers to
workers, a sustainable equilibrium can be reached due to lower average
depletion per capita — an equivalent δ if everyone contributed equally
to labor. This could also be interpreted as modeling a reduction in the average workload per worker.
5.3. Unequal Society (with Elites and Commoners): xE ≥ 0 and κ N 1
In our examples of an unequal society, the Elites (per capita) consume κ b 10 to 100 times more than the Commoners. Their population,
plotted in red, is multiplied by κ to represent their equivalent impact because of their higher consumption. That is why we use the label “Equivalent Elites” on the graphs in this Section 5.3.
In the first two cases, we discuss two distinct, but generic types
of collapse in an unequal society. In these two scenarios, κ = 100.
Then we will show possibility of reaching an equilibrium by reducing κ to 10 and adjusting the birth rates βE and βC independently. These
two κ = 10 scenarios show that in order to reach a sustainable equilibrium in an unequal society, it is necessary to have policies that limit inequality and ensure birth rates remain below critical levels.
5.3.1. Unequal Society: Type-L Collapse (Labor Disappears, Nature
Recovers)
This scenario, presented in Fig. 6a, is precisely the same as the equilibrium without Elites case presented in Section 5.1.1 (Fig. 3a) except
that here we set xE(0) = 1.0 × 10−3. This is indeed a very small initial
seed of Elites. The two scenarios look pretty much the same up until
about t = 500 years after the starting time of the simulation. The Elite
population starts growing significantly only after t = 500, hence depleting the Wealth and causing the system to collapse. Under this scenario,
the system collapses due to worker scarcity even though natural resources are still abundant, but because the depletion rate is optimal, it
takes more than 400 years after the Wealth reaches a maximum for the
society to collapse. In this example, Commoners die out first and Elites
disappear later. This scenario shows that in a society that is otherwise sustainable, the highly unequal consumption of elites will still cause a
collapse.
This scenario is an example of a Type-L collapse in which both Population and Wealth collapse but Nature recovers (to its maximum capacity,
λ, in the absence of depletion). Scarcity of workers is the initial cause of a
Type-L collapse, as opposed to scarcity of Nature for a Type-N collapse.
5.3.2. Unequal Society: Irreversible Type-N Collapse (Full Collapse)
The typical scenario in Fig. 6b for a full collapse is the result of running the model with the parameter values and initial conditions given
by Table 1. Examples of irreversible Type-N (full) collapses in the egalitarian and equitable societies are presented in Section 5.1.4 (Fig. 3d) and
Section 5.2.4 (Fig. 4d).
We set a small initial seed of xE(0) = 0.20, κ = 100, and a large
depletion δ = 1.0 × 10−4, so that both the depletion δ = 15δ* and the inequality coefficient κ = 100 are very large. This combination results in a
full collapse of the system with no recovery. The Wealth starts declining
as soon as the Commoner's population goes beyond its carrying capacity,
and then the full collapse takes only about 250 additional years. The declining Wealth causes the fall of the Commoner's population (workers)
with a time lag. The fast reduction in the number of workers combined
with scarcity of natural resources causes the Wealth to decline even faster
than before. As a result, the Elites – who could initially survive the famine
due to their unequal access to consumable goods (κ = 100) – eventually
also die of hunger. Note that because both depletion and inequality are
large, the collapse takes place faster and at a much lower level of population than in the previous case (see Section 5.3.1, Fig. 5.3.1) with a
depletion rate of δ = δ*.
99
5.3.3. Unequal Society: Soft Landing to Optimal Equilibrium
The following parameter values and initial values can produce the
current scenario (the rest are exactly the same as in Table 1):
8
>
< βC
xC ð0Þ
>
:
κ
¼
¼
¼
−2
6:5 10
þ4
1:0 10
10
βE
xE ð0Þ
δ
−2
¼ 2:0 10
þ3
:
¼ 3:0 10
−6
¼ 6:35 10
ð20Þ
The value for δ used in this scenario is δ*** given by Eq. (18). It must
be remembered that ψ = 0.65 is not a parameter that we can choose.
However, it can be read from the result of the simulation since it is the
equilibrium ratio of the Elite to Commoner population. See the second
equation in Eq. (17). On the other hand, η ¼ 121 is determined by the
death and birth rates as well as the inequality coefficient. These parameters are chosen in order to satisfy Eq. (16), the necessary condition for
attaining an equilibrium in an unequal society.
The same comments as in Section 5.1.1 hold here if we choose a
lower-than-optimal δ.
5.3.4. Unequal Society: Oscillatory Approach to Equilibrium
The parameter values and initial conditions in the scenario presented in Fig. 6d are exactly the same as the previous scenario, presented in
Fig. 6c, except for δ. It is increased to 1.3 × 10−5, almost 2δ***. This results
in a much lower Carrying Capacity compared to 5.3.3, as can be seen
from a comparison of Fig. 6c and d. Therefore, the total final population
in the present scenario is much less than the total final population in the
previous scenario, 5.3.3 (Fig. 6c) (Table 2).
6. Discussion of Results
We conducted a series of experiments with the HANDY model, considering first an egalitarian society without Elites (xE = 0), next an equitable society (κ = 1) where Non-Workers and Workers are equally paid,
and finally an unequal society whose Elites consume κ times more
than the Commoners. The model was also used to find a sustainable
equilibrium value and the maximum carrying capacity within each of
these three types of societies.
6.1. Unequal Society
The scenarios most closely reflecting the reality of our world today
are found in the third group of experiments (see the scenarios for an unequal society in Section 5.3), where we introduced economic stratification. Under such conditions, we find that collapse is difficult to avoid,
which helps to explain why economic stratification is one of the elements recurrently found in past collapsed societies. Importantly, in
the first of these unequal society scenarios, 5.3.1, the solution
appears to be on a sustainable path for quite a long time, but even using
an optimal depletion rate (δ*) and starting with a very small number
of Elites, the Elites eventually consume too much, resulting in a famine
among Commoners that eventually causes the collapse of society. It is
important to note that this Type-L collapse is due to an inequalityinduced famine that causes a loss of workers, rather than a collapse of
Nature. Despite appearing initially to be the same as the sustainable
optimal solution obtained in the absence of Elites, economic stratification
changes the final result: Elites' consumption keeps growing until the society collapses. The Mayan collapse – in which population never recovered
even though nature did recover – is an example of a Type-L collapse,
whereas the collapses in the Easter Island and the Fertile Crescent –
where nature was depleted – are examples of a Type-N collapse.
In scenario 5.3.2, with a larger depletion rate, the decline of the
Commoners occurs faster, while the Elites are still thriving, but eventually the Commoners collapse completely, followed by the Elites. It is important to note that in both of these scenarios, the Elites – due to their
wealth – do not suffer the detrimental effects of the environmental
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collapse until much later than the Commoners. This buffer of wealth allows Elites to continue “business as usual” despite the impending catastrophe. It is likely that this is an important mechanism that would help
explain how historical collapses were allowed to occur by elites who appear to be oblivious to the catastrophic trajectory (most clearly apparent in the Roman and Mayan cases). This buffer effect is further
reinforced by the long, apparently sustainable trajectory prior to the beginning of the collapse. While some members of society might raise the
alarm that the system is moving towards an impending collapse and
therefore advocate structural changes to society in order to avoid it,
Elites and their supporters, who opposed making these changes, could
point to the long sustainable trajectory “so far” in support of doing
nothing.
The final two scenarios in this set of experiments, 5.3.3 and 5.3.4, are
designed to indicate the kinds of policies needed to avoid this catastrophic outcome. They show that, in the context of economic stratification, inequality must be greatly reduced and population growth must be
maintained below critical levels in order to avoid a societal collapse
(Daly, 2008).
depletion parameter further results in cycles of large overshooting,
major collapses, and then eventual recovery of Nature. Scenario 5.2.4
shows us that increasing depletion per capita further can produce an irreversible Type-N collapse.
Finally, scenario 5.2.5, which is a replication of the scenario in
5.2.4 with a much higher ratio of Non-Workers to Workers, shows
that a collapse in an equitable society could be avoided by reducing
the average depletion per capita. We note that this scenario could also
represent a situation where, rather than having paid Non-Workers,
the workload per capita is reduced, with the whole population working
“fewer days a week”. Such a “work-sharing” policy has been successfully
implemented in Germany over the past few years for reducing unemployment (Baker and Hasset, 2012; Hasset, 2009). Moreover, Knight
et al. (2013) show, through a panel analysis of data for 29 highincome OECD countries from 1970 to 2010, that reducing work hours
can contribute to sustainability by reducing ecological strain. This conclusion agrees with our comparison of the two scenarios, 5.2.5 and
5.2.4, presented above.
6.2. Egalitarian Society
6.4. HANDY and Brander–Taylor Model
In order to further understand what conditions are needed to avoid
collapse, our first set of experiments model a society without economic
stratification and start with parameter values that make it possible to
reach a maximum carrying capacity (scenario 5.1.1). The results
show that in the absence of Elites, if the depletion per capita is kept at
the optimal level of δ*, the population grows smoothly and asymptotes
the level of the maximum carrying capacity. This produces a softlanding to equilibrium at the maximum sustainable population and production levels.
Increasing the depletion factor slightly (scenario 5.1.2) causes the
system to oscillate, but still reach a sustainable equilibrium, although,
importantly, at a lower carrying capacity. Population overshoots its carrying capacity, but since the overshoot is not by too much – of the order
of the carrying capacity – the population experiences smaller collapses
that can cause it to oscillate and eventually converge to a sustainable
equilibrium. Thus, while social disruption and deaths would occur, a
total collapse is avoided.
A further increase in the depletion factor (scenario 5.1.3) makes the
system experience oscillatory periods of growth, very large overshoots
and devastating collapses that almost wipe out society, but the eventual
recovery of Nature allows for the cycle to be repeated.
Increasing the depletion factor even further (scenario 5.1.4) results
in a complete collapse of the system. This shows that depletion alone,
if large enough, can result in a collapse — even in the absence of economic stratification.
As previously mentioned, a similar use of the predator–prey approach was applied in the pioneering work of Brander and Taylor
(1998) (BT) to study the historical rise and fall of the Easter Island population. In comparison to their model, with just two equations for
Population and Nature, the introduction of Elites and Commoners,
and accumulated Wealth, results in a greater variety and broader spectrum of potential solutions. Moreover, the collapse scenario presented
in BT is somewhat different from the ones presented above. As a matter
of fact, the collapse scenario presented in Fig. 3 of Brander and Taylor
(1998) seems to be more of an oscillatory approach to equilibrium, similar to the one shown in our Fig. 3b, and not a collapse in the sense that
we define in this paper. Furthermore, the carrying capacity, in the sense
we define in this paper, is also different from what BT (1998) call carrying capacity. Indeed, their carrying capacity (K) is our Nature's capacity,
λ, which is the maximum size Nature can reach, whereas Carrying Capacity in HANDY is the population level that can be supported by a
given level of natural resources. Furthermore, BT's carrying capacity is
a constant, whereas Carrying Capacity in HANDY adjusts according to
the level of depletion of Nature.
While sharing certain similarities with the Brander and Taylor
model, our more complex model structure and the use of different assumptions, allows our model to apply to multiple types of societies
with varying socioeconomic structures. Thus, unlike works that tend
to study further implications of the two-dimensional model of BT
(Anderies, 2000), the model we have developed introduces a more
complex set of possible feedbacks and nonlinear dynamics, and a greater spectrum of potential outcomes. This allows HANDY to model a different and wider set of thought experiments.
An important feature of HANDY that distinguishes it from predator–
prey, BT, and other similar models (Anderies, 1998; Dalton et al., 2005;
Erickson and Gowdy, 2000; Reuveny and Decker, 2000) is its native capability for producing irreversible collapses due to the structure for accumulation of wealth. Our approach also differs from models like
D'Alessandro (2007) that can produce irreversible collapses but only
through explicit introduction of a critical depensation mechanism into
the model. The dynamics produced by HANDY offer the possibility of irreversible collapses without having to introduce such an additional
mechanism into the model. See Section 5.1.4 for an explanation of irreversible collapses in HANDY.1
6.3. Equitable Society (with Workers and Non-Workers)
As the second set of experiments (presented in Section 5.2) show,
HANDY allows us to model a diverse range of societal arrangements.
In this set of experiments, choosing xE ≥ 0 and κ = 1 has allowed us
to model a situation that can be described as having Workers and
Non-Workers with the same level of consumption, i.e., with no economic stratification. The Non-Workers in these scenarios could represent a
range of societal roles from students, retirees, and disabled people, to intellectuals, managers, and other non-productive sectors. In this case, the
Workers have to deplete enough of Nature to support both the NonWorkers and themselves.
The first scenario, 5.2.1, shows that even with a population of NonWorkers, the total population can still reach a sustainable equilibrium
without a collapse. In scenario 5.2.2, we find that increasing the depletion factor induces a series of overshoots and small collapses where population eventually converges to a lower sustainable equilibrium. Like in
an egalitarian society, scenario 5.2.3 shows us that increasing the
1
We wish to acknowledge and thank reviewer No. 1 for highlighting these very important points to us.
S. Motesharrei et al. / Ecological Economics 101 (2014) 90–102
7. Summary
Collapses of even advanced civilizations have occurred many times
in the past five thousand years, and they were frequently followed by
centuries of population and cultural decline and economic regression.
Although many different causes have been offered to explain individual
collapses, it is still necessary to develop a more general explanation. In
this paper we attempt to build a simple mathematical model to explore
the essential dynamics of interaction between population and natural
resources. It allows for the two features that seem to appear across societies that have collapsed: the stretching of resources due to strain
placed on the ecological carrying capacity, and the division of society
into Elites (rich) and Commoners (poor).
The Human And Nature DYnamical model (HANDY) was inspired by
the predator and prey model, with the human population acting as the
predator and nature being the prey. When small, Nature grows exponentially with a regeneration coefficient γ, but it saturates at a maximum value λ. As a result, the maximum regeneration of nature takes
place at λ / 2, not at the saturation level λ. The Commoners produce
wealth at a per capita depletion rate δ, and the depletion is also proportional to the amount of nature available. This production is saved as accumulated wealth, which is used by the Elites to pay the Commoners a
subsistence salary, s, and pay themselves κs, where κ is the inequality
coefficient. The populations of Elites and Commoners grow with a
birth rate β and die with a death rate α which remains at a healthy
low level when there is enough accumulated food (wealth). However,
when the population increases and the wealth declines, the death rate
increases up to a famine level, leading to population decline.
We show how the carrying capacity – the population that can be indefinitely supported by a given environment (Catton, 1980) – can be
defined within HANDY, as the population whose total consumption is
at a level that equals what nature can regenerate. Since the regrowth
of Nature is maximum when y = λ / 2, we can find the optimal level
of depletion (production) per capita, δ* in an egalitarian society where
xE ≡ 0, δ∗∗(≥ δ∗) in an equitable society where κ ≡ 1, and δ*** in an unequal society where xE ≥ 0 and κ N 1.
In sum, the results of our experiments, discussed in Section 6, indicate that either one of the two features apparent in historical societal
collapses – over-exploitation of natural resources and strong economic
stratification – can independently result in a complete collapse. Given
economic stratification, collapse is very difficult to avoid and requires
major policy changes, including major reductions in inequality and population growth rates. Even in the absence of economic stratification, collapse can still occur if depletion per capita is too high. However, collapse
can be avoided and population can reach equilibrium if the per capita
rate of depletion of nature is reduced to a sustainable level, and if resources are distributed in a reasonably equitable fashion.
In the upcoming generations of HANDY, we plan to develop several
extensions including: (1) disaggregation of Nature into nonrenewable
stocks, regenerating stocks, and renewable flows, as well as the introduction of an investment mechanism in accessibility of natural resources, in order to study the effects of investment in technology on
resource choice and production efficiency; (2) making inequality (κ)
endogenous to the model structure; (3) introduction of “policies” that
can modify parameters such as depletion, the coefficient of inequality,
and the birth rate; and, (4) introduction of multiple coupled regions to
represent countries with different policies, trade of carrying capacity,
and resource wars.
Those interested in obtaining the model code can contact the authors.
Acknowledgments
We are grateful to Profs. Matthias Ruth, Victor Yakovenko, Herman
Daly, Takemasa Miyoshi, Jim Carton, Fernando Miralles-Wilhelm, and
Ning Zeng, and Drs. Robert Cahalan and Steve Penny for many useful
discussions. Study of the “Equitable Society” scenarios (i.e., with
101
Workers and Non-Workers), the scenario presented in Section 5.2.5,
in particular, was suggested by V. Yakovenko. We would also like to
thank anonymous reviewer No. 1 for having highlighted to us the importance of the capability of HANDY to naturally produce irreversible
collapses, which is not found in earlier models. We would especially
like to thank the editors of this journal for alerting us to the model
and work done by Brander and Taylor, of which we were unaware,
and allowing us to revise our article to account for this new information.
This work was partially funded through NASA/GSFC grant NNX12AD03A.
Based on the media reports on a pre-publication version of this paper,
NASA issued the official statement contained in Release 14-082:
http://www.nasa.gov/press/2014/march/nasa-statement-onsustainability-study/
March 20th, 2014
RELEASE 14-082
NASA Statement on Sustainability Study
The following is a statement from NASA regarding erroneous media
reports crediting the agency with an academic paper on population
and societal impacts.
"A soon-to-be published research paper 'Human and Nature Dynamics (HANDY): Modeling Inequality and Use of Resources in the Collapse or Sustainability of Societies' by University of Maryland
researchers Safa Motesharrei and Eugenia Kalnay, and University of
Minnesota’s Jorge Rivas was not solicited, directed or reviewed by
NASA. It is an independent study by the university researchers utilizing research tools developed for a separate NASA activity."
"As is the case with all independent research, the views and conclusions in the paper are those of the authors alone. NASA does not endorse the paper or its conclusions."
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